Submission date: 3 July 2015.

Abstract:

Starting with a differentially closed field F =(F ; + ,⋅ , 0 , 1 , D) we consider reducts F_E=(F ; + , ⋅ , 0 , 1 , E) with a binary relation E(x,y) for a differential algebraic curve given by a differential polynomial equation f(x,y)=0. The main problem is to understand for which differential curves the derivation D of F is definable in F_E. We present several classes of curves for which the derivation is definable and also give some necessary and sufficient conditions for definability of D. As a consequence we obtain a characterisation of definable and algebraic closures in the reducts.

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Full text arXiv 1507.00971: pdf, ps.